Fit Log-Linear Models by Iterative Proportional Scaling

This function provides a front-end to the standard function, loglin, to allow log-linear models to be specified and fitted in a manner similar to that of other fitting functions, such as glm.

models, category
loglm(formula, data, subset, na.action, ...)
A linear model formula specifying the log-linear model.

If the left-hand side is empty, the data argument is required and must be a (complete) array of frequencies. In this case the variables on the right-hand side may be the na

Numeric array or data frame. In the first case it specifies the array of frequencies; in then second it provides the data frame from which the variables occurring in the formula are preferentially obtained in the usual way. This argu
Specifies a subset of the rows in the data frame to be used. The default is to take all rows.
Specifies a method for handling missing observations. The default is to fail if missing values are present.
May supply other arguments to the function loglm1.

If the left-hand side of the formula is empty the data argument supplies the frequency array and the right-hand side of the formula is used to construct the list of fixed faces as required by loglin. Structural zeros may be specified by giving a start argument with those entries set to zero, as described in the help information for loglin.

If the left-hand side is not empty, all variables on the right-hand side are regarded as classifying factors and an array of frequencies is constructed. If some cells in the complete array are not specified they are treated as structural zeros. The right-hand side of the formula is again used to construct the list of faces on which the observed and fitted totals must agree, as required by loglin. Hence terms such as a:b, a*b and a/b are all equivalent.


  • An object of class "loglm" conveying the results of the fitted log-linear model. Methods exist for the generic functions print, summary, deviance, fitted, coef, resid, anova and update, which perform the expected tasks. Only log-likelihood ratio tests are allowed using anova.

    The deviance is simply an alternative name for the log-likelihood ratio statistic for testing the current model within a saturated model, in accordance with standard usage in generalized linear models.


If structural zeros are present, the calculation of degrees of freedom may not be correct. loglin itself takes no action to allow for structural zeros. loglm deducts one degree of freedom for each structural zero, but cannot make allowance for gains in error degrees of freedom due to loss of dimension in the model space. (This would require checking the rank of the model matrix, but since iterative proportional scaling methods are developed largely to avoid constructing the model matrix explicitly, the computation is at least difficult.) When structural zeros (or zero fitted values) are present the estimated coefficients will not be available due to infinite estimates. The deviances will normally continue to be correct, though.


Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

See Also

loglm1, loglin

  • loglm
# The data frames  Cars93, minn38 and quine are available
# in the MASS package.

# Case 1: frequencies specified as an array.
sapply(minn38, function(x) length(levels(x)))
## hs phs fol sex f
##  3   4   7   2 0
minn38a <- array(0, c(3,4,7,2), lapply(minn38[, -5], levels))
minn38a[data.matrix(minn38[,-5])] <- minn38$fol
fm <- loglm(~1 + 2 + 3 + 4, minn38a)  # numerals as names.
##[1] 3711.9
fm1 <- update(fm, .~.^2)
fm2 <- update(fm, .~.^3, print = TRUE)
## 5 iterations: deviation 0.0750732
anova(fm, fm1, fm2)
LR tests for hierarchical log-linear models

Model 1:
  ~  1 + 2 + 3 + 4
Model 2:
 .  ~  1 + 2 + 3 + 4 + 1:2 + 1:3 + 1:4 + 2:3 + 2:4 + 3:4
Model 3:
 .  ~  1 + 2 + 3 + 4 + 1:2 + 1:3 + 1:4 + 2:3 + 2:4 + 3:4 +
        1:2:3 + 1:2:4 + 1:3:4 + 2:3:4

          Deviance  df Delta(Dev) Delta(df) P(> Delta(Dev)
  Model 1 3711.915 155
  Model 2  220.043 108   3491.873        47        0.00000
  Model 3   47.745  36    172.298        72        0.00000
Saturated    0.000   0     47.745        36        0.09114
# Case 1. An array generated with xtabs.

loglm(~ Type + Origin, xtabs(~ Type + Origin, Cars93))
loglm(formula = ~Type + Origin, data = xtabs(~Type + Origin,

                    X^2 df  P(> X^2)
Likelihood Ratio 18.362  5 0.0025255
         Pearson 14.080  5 0.0151101
# Case 2.  Frequencies given as a vector in a data frame
## [1] "Eth"  "Sex"  "Age"  "Lrn"  "Days"
fm <- loglm(Days ~ .^2, quine)
gm <- glm(Days ~ .^2, poisson, quine)  # check glm.
c(deviance(fm), deviance(gm))          # deviances agree
## [1] 1368.7 1368.7
c(fm$df, gm$df)                        # resid df do not!
c(fm$df, gm$df.residual)               # resid df do not!
## [1] 127 128
# The loglm residual degrees of freedom is wrong because of
# a non-detectable redundancy in the model matrix.
Documentation reproduced from package MASS, version 7.3-3, License: GPL-2 | GPL-3

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